In this appendix, we show that Schwinger representation of the simplex integral () is given by (REF ) with graph polynomials given in (REF ). Our discussion builds on that in [1]}.
Labelling the vertices of the simplex by \(i=1,\ldots ,n\) , and the (directed) legs by \(a=1,\ldots ,N\)  where \(N=n(n-1)/2\) , we introduce the incidence matrix
\(\epsilon _i^a = {\left\lbrace \begin{array}{ll} +1\qquad &\mathrm {if\, leg}\, a\, \mathrm {is\, ingoing\, to\, vertex}\, i\\-1\qquad & \mathrm {if\, leg}\, a\, \mathrm {is\, outgoing\, to\, vertex}\, i\\0\qquad &\mathrm {otherwise}\end{array}\right.}\) 
